The linear natural convection of a Maxwell viscoelastic fluid with Cattaneo–Christov heat flux constitutive equation, between two thick walls with finite thermal conductivity is investigated. The viscoelastic fluid and the heat flux have different relaxation times. The main interest is on the curves of criticality for different thicknesses ratio D and thermal conductivities ratio X. In the middle range of log X the curves of criticality stabilize depending on the other parameters of the problem. It is revealed that for some Prandtl numbers the curves of criticality remain higher for small X and lower for large X. However, increasing the Prandtl number this behavior is reversed. It is shown that this has important consequences in the order of the criticality curves when the heat flux relaxation time is increased. Depending on the Prandtl number, an increase of this relaxation time may decrease (destabilize) the curves of criticality until a minimum is reached, after which the curves start to increase (stabilize) again. For two different magnitudes of the viscoelastic relaxation time, the critical Rayleigh number, wavenumber and frequency of oscillation are plotted against log X for different magnitudes of D and the heat flux relaxation time.